The generator matrix

 1  0  0  1  1  1  X  1  1  0  1 X^3  X  1  1  1  1 X^3+X^2  1 X^3 X^2+X  1  1  1 X^2  X  X  0 X^3+X^2+X  1  1  1 X^3+X^2 X^3+X^2+X X^2+X X^3+X^2  1  1 X^3  1  1 X^3  1  1  1  1 X^2+X X^3+X  1  1  X  1  1
 0  1  0  0 X^2+1 X^3+X+1  1 X^3+X^2+X+1 X^2+X X^2 X^3+X+1  1  1 X^2 X^3+X^2+X  1 X^3+X^2+1 X^3+X^2+X  X  1  1 X^3 X^2+X+1 X^3+X^2+X  1 X^3+X  1  1 X^3+X  1 X^3+X+1 X^3  1  1  1  1  1 X^3+X+1  1 X^3+X^2 X^3+X^2+1  1  1 X^3+X^2+X+1 X^3+X^2 X^2+1  1  1 X+1 X^3+1  0 X^3+1 X^2
 0  0  1  1  1 X^2 X^2+1 X^3+X+1 X^3+1  1 X^3+X X^3+X^2 X^3+1 X^3+X^2+X X^3+1 X^3+X^2 X^2+X+1  1  0 X^3+1  0  X X+1 X^3+X^2+X+1 X^3+X^2+1  1 X^3+X^2+X X^3+X+1  1  X  1 X^3+X^2+1 X^2+X X^3  X X^3+X^2 X^3+X X^3+X^2+X X^3+1 X^3+X^2+X+1  0 X+1 X^2+X+1 X^2 X^2 X^2+1 X^3  X X+1 X^2+X  1 X^3+X^2+X+1  0
 0  0  0  X X^3+X X^3 X^3+X X^3+X^2+X  X  X X^2 X^3+X X^2 X^2+X X^3+X^2 X^3+X^2+X X^3+X^2 X^3+X^2+X X^3+X X^2+X X^2  0  0  0 X^3+X^2 X^3+X^2+X X^2+X X^3+X^2+X X^3 X^3+X X^2+X X^3+X^2 X^3  0 X^3+X^2 X^2+X X^3 X^3+X^2+X X^3 X^3+X^2 X^3+X^2 X^2 X^2+X X^3+X X^2 X^3+X^2+X X^3+X^2+X X^3 X^3 X^3+X^2 X^3 X^3+X^2+X X^2

generates a code of length 53 over Z2[X]/(X^4) who´s minimum homogenous weight is 47.

Homogenous weight enumerator: w(x)=1x^0+156x^47+926x^48+1624x^49+2932x^50+3896x^51+4537x^52+5246x^53+4472x^54+3428x^55+2575x^56+1610x^57+890x^58+236x^59+127x^60+46x^61+49x^62+12x^63+2x^64+2x^65+1x^70

The gray image is a linear code over GF(2) with n=424, k=15 and d=188.
This code was found by Heurico 1.16 in 9.53 seconds.